THE EFFECTS OF CORNER RAD11 ON THE LOCAL

BUCKLING OF COLD-F'ORMED SECTIONS

ARASH NASSIRIRAD

A Thesis

in

SCWOOL FOR BUILDIlVG

Presented in partial fulfillrnent of the requirements

for the degree of Master of Applied Sciences (Building) at

Concordia Unimrsity.

Montreal, Quebec, Canada

September 1998

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ABSTRACT

THE EFFECTS OF CORNER RADII ON THE LOCAL BUCKltING OF

COLD-FORMED SECTIONS

ARASH NASSIRIRAD

In steel construction the methods to produce structural members include hot

rolling, forming and welding hollow section (HSS) and welding plates to

form 1's (WWF). Another method, which is less widely known but is

growing in acceptance, is cold-forming fiom steel sheet or strip using roll-

forming machines, press and bending brakes, or folding operations. These

cold-forrn steel sections have their own design codes.

The use of cold-fonned structures in building constructions goes back to

about the 1850s in both the United States and Great Britain but the forma1

design methods for these sections were developed only since 1940.

Cold-formed sections add aspects to the design procedure which are

neglected in codes for hot-rolled shapes, namely local and torsional

buckling. The elements of a box or channel section formed from thin sheet

may buckle locally, leading to collapse. To calculate the buckling stress the

iii

manufacturing process, makes the measurement of the width uncertain.

Codes simply adopt the flat width. The objective of this study is to develop a

more accurate method to predict the buckling stresses for two major kinds of

cold-formed section, namely an open section (channel) and a closed

section (box).

A theory has been developed to mode1 the elastic local buckling mode for

box and channel sections. This is extended to predict the behavior of the

section after buckling and up to the collapse. Cornparisons are made with the

results of the numeric analysis carried out using the finite element program

ABAQUS, and with the code requirements.

To Dr. Shahin Izadian

who supports this study by her life and spirit.

ACKNOWLEDGrnNTS

The author wishes to express his gratitude to ProfessorCedric Marsh, for the

initial encouragement in the project, and for his continuous guidance and

suggestions, which have enriched the content of this work.

The author is also grateful to his colleagues, for their valuable criticism,

helpful advice and solid support.

Financial support fiom NSERC is also acknowledged.

TABLE OF CONTENTS

LIST OF FIGURES

LIST OF TABLES

LIST OF SYMBOLS

DEFINITIONS

CHAPTER 1

INTRODUCTION

1.1 General

1.2 Behavior of cold formed members

1 -2.1 Rectangular box columns

1.2.2 Channel columns

CHAPTER 2

THEORETICAL MODEL

2.2 Overall buckling

PAGE

xi

xiv

xv

xix

vii

2-2 Box section

2-2-1 Code procedures

2.2.2 Proposed analytical model

2.2.3 Theoretical post-buckling

2.3 Channel section

2.3.1 Code procedures

2.3.2 Proposed analytical model

2.3.4 Theoretical post-buckling

CHAPTER 3

FINITE ELEMENT STUDIES

3.1 Introduction

3.2 Eigenvalue method

3.3 Ultimate load

3.4 ABAQUS program

3.4.1 Introduction

3.4.2 Input of the program

3 -4.3 Output of the program

viii

CHAPTER 4

COMPARJNG THE RESULTS

4.1 Typical example

4.1.1 Theoretical calculation

4.1.2 Code predictions

4.1.3 Finite element results

4.1.3.1 Eigenvalue result

4.1.3.2 Postbuckling strength

4.2 Results for box sections

4.3 Results for channel sections

4.4 Conclusions

4.4.1 Box section

4.4.2 Channel section

4.4.3 Final comments

4.4.4 Future research

APPENDIX 1

ANALYSIS

1.1 Introduction

1.2 Box section

1.3 Channel section

1.3.1 Flange mode

1.3.2 Web mode

APPErnIX II

WARPING CONSTANT

11.1 Box section

11.2 Channel section

APPENDIX III

List 1

List 2

FIGURES

TABLES

REFERENCES

1-1

1-9

1-9

1-1 3

II

II- 1

II- 1

11-4

III

III- I

111-4

F1

T l

R1

LIST OF FIGURES

Figure D-1 Local buckling for a box section

Figure D-2 Local buckling for a channel section

Figure D-3 Flat, overall and effective widths

Figure D-4 Collapse mode for a box section

Figure D-5 Warping for 1 beam

Figure 1 - 1 Cold-formed sections

Figure 1-2 Cold-forrned steel structure

Figure 1-3 Cold-formed and hot-rolled steel structure

Figure 1-4 Typical cold-formed panels

Figure 1-5 Force-shortening relationship for column and plate

Figure 1-6 Box and panel section dimensions

Figure 1-7 Stress distribution on a simply supported plate

Figure 2-1 Buckling and postbuckling strength

Figure 2-2 Cold-formed and hot-rolled corners

Figure 2-3 Simply Supported plate

Figure 2-4 Postbuckling curve

Figure 2-5 Buckling modes of a channel

Figure 3- 1 Sarnple plate for ABAQUS

FD- 1

FD-2

FD-3

FD-4

FD-5

f1-1

f1-2

f1-3

F 1-4

f1-5

F1-6

Figure 4-1 Typical box section

Figure 4-2 ABAQUS elastic buckling

Figure 4-3 Stress distribution across the wall of a box section

Figure 4-4 Mean stress and axial shorting for a box section

Figure 4-5 to 4-34 Critical and ultimate stress for box sections

Figure 4-35 to 4-37 m factor for box sections

Figure 4-38 to 4-40 Normalized m for box sections

Figure 4-4 1 Normalized mean postbuckling strength for box sections

Figure 4-42 to 4-56 Critical strsss for channel sections

Figure 4-57 to 4-59 m factor for channei sections

Figure 4-60 to 4-62 Nomalized m for charme1 sections

Figure 4-63 Normalized mean strength for channel sections

Figure 1-1 Deflection of a box section

Figure 1-2 Deflection of a channel section (flange mode)

Figure 1-3 Deflection of a channel section (web mode)

Figure 11-1 Corner of a box section

Figure 11-2 Warping constant construction for the corner of a box section

Figure 11-3 Warping constant for box sections

F4- 1

F4-2

F4-3

F4-4

F4-5 to F4-34

F4-35 to f4-37

F4-38 to F4-40

F4-4 1

F4-42 to F4-56

F4-57 to f4-59

F4-60 to F4-62

F4-63

FA- 1

FA-%

FA-3

FA-4

FA-5

FA-6

xii

Figure 11-4 Warping factor a for box sections FA-7

Figure 11-5 Corner of a channel section FA-8

Figure 11-6 Warping constant construction for the corner of a charme1 section FA-9

Figure 11-7 Warping constant for channel sections FA-10

Figure 11-8 Warping factor a for channel sections FA-1 1

xiii

LIST OF TABLES

Table 4- 1 Summary of results for box sections

Table 4-2 Sumrnary of results for channel sections

xiv

LIST OF SYMBOLS

Unreduced cross-sectional area of section

Effective cross-sectional area of section

Plate rigidity

Elastic modulus

Elastic buckling force

Ultimate force

Ultimate load from ABAQUS

Warping constant

Factor

Elastic stiffness matrix

Initial stiffhess matrix

Length

Bending moment

Axial flux

Loading pattern matrix

Bend radius to median line

Strain energy due to bending

Strain energy due to warping

Strain energy due to bending of web

External work

Work done by applied stresses

Axis

Axis

Axis

Shorter width of the box and width of the web of a channel

Deflection coefficients

Longer width of the box and width of the flange of a channel

Effective width of the box and flange of a channel

Halfwave length of a buckied element

Distance to shear centre

Factor

Normalized m factor

True length of section

Thickness

Displacement in X direction

xvi

Increment in displacement for load-displacement control

Displacement in Y direction

Flat width

Flat width limit

Distance along X axis

Distance along Y axis

Distance along Z axis

Warping factor

Angle of rotation

Factor

Deflection matrix for eigenvalue

Angle of twist at the corner

Buckling mode matrix for eigenvalue

Slenderness

Normalized slenderness

Load multipliers for eigenvalue

Poisson ratio (0.3 for steel)

Stress

xvii

ABAQUS o a v

ABAQ US O c r

Normalized stress

Actual buckling stress

Elastic buckling stress

Actual buckling stress for web of channel section

Yield strength

Average ultimate stress fkom ABAQUS

Elastic critical stress fiom ABAQUS

xviii

DEFINITIONS

LOCAL BUCKLING shown in Fig D-1 for box and Fig D-2 for a

channel is a wavelike deflection form, with a wavelength

independent of the overall length.

CORNER RADIUS: is the radius of the median line at the corners

of a formed section. It is usually a multiple of the thickness.

CRITICAL FORCE (STRESS): is the force (stress) which causes

initial elastic buckling of a member or element.

EFFECTIVE DESIGN WIDTH (b,,): is the width assumed to carry

the yield strength, used to calculate the collapse load.

FLAT WIDTH (w): is the width of the straight portion of the

element, Fig D-3.

LZMITING WIDTH/THI:CKNESS RATIO (w/t): is the

widWthickness ratio below which local buckling need not be

considered.

OVERALL WIDTH (b): is the overall width of the element

between points of intersections of the median lines, Fig D-3.

xix

ULTIMATE FORCE: is the load which causes the member to

collapse, as shown in Fig D-4

WlDTH/THICKNESS RATIO: is either the "flat width ratio"

which is the ratio of the flat width (w) to the thickness (t) or

bboverall width ratio" which is the ratio of the overall width (b) to

the thickness. (Fig D-3)

0 WARPING: when twisting in open sections causes translation in the

planes of elements of the section, the cross sections tend to warp

from the original flat plane, as shown inFig D-5. If this warping is

resisted it contributes to the torsional stiffhess.

m YIELD: is the stress at the limit of the elastic range, or at a 0.2%

permanent offset strain.

CHAPTER 1

INTRODUCTION

1.1 GENIEIRAL,

Cold-formed steel sections represent an alternative to hot rolled sections in

applications where the light weight sections can be used.

Although the use of cold-formed sections in construction has been known

since the middle of the last century the use of cold formed members in

buildings was fonnalized with the publication of "Specification for the

Design of Cold-Forrned Steel Structural Members" by the American Iron

and Steel Institute (AISI 1946) which was the first to provide a fomal design

procedure base on the seminal work of Winter (1937 ). The most recent AIS1

code is "Specification for the design of cold-fomed steel structural

mernbers" (1996). The first Canadian standard, "The Design of Light Gauge

Steel Structural Members" appeared in 1963, while the latest CSA-S 136-

M94 "Cold formed steel structural members" was issued in 1994.

The Arnerica Society of Civil Engineers (ASCE) prepared a standard

ANSYASCE 8-90 "Specification for the Design of Cold-Formed Stainless

Steel Structura1 Members" in 1990.

By using cold foming it is possible to produce more efficient structural

shapes, which have been used extensively in airplanes and car bodies, but in

the construction industry the use was limited to roofing sheet and panels.

Over the past 40 years, several studies for the design of thin walled sections

have led to an increasing use of cold-formed steel in building structures.

Open sections used as wall studs, floor and roof beams are found in

institutional, commercial , residential and light factory buildings. Storage

racks and latticed joists incorporate special cold fonned shapes.

The major difference between cold-formed sections and hot-rolled sections

is the relative thickness, or (b/t) ratio, which leads to the following

advantages:

Cold-formed members c m be manufactured more easily.

Unusual sectional configurations can be produced economically and

consequently, more favorable strength-to-weight ratios can be

obtained.

Nesting sections can be produced, allowing for compact packaging

for shipping.

Great accuracy of the profile

r A wider variety of shapes

A wider variety of steel tempers

Coated and painted sheet

a Sections c m be perforated

Cold-formed steel structural products can be classified into two major types:

Individual structural fiaming members.

m Panels and decks.

Fig 1-1 shows some of the cold-formed sections generally used in structural

fiarning. The more usual shapes are channels, 2-sections, angles, hat

sections.

In general the thickness of cold-formed rnembers ranges fkom 1 to 10 mm.

In view of the fact that the major function of this type of individual h i n g

member is to cany load, structural strength and stiffhess are the main

considerations in design. Such sections can be used as primary fiaming

members in buildings. Fig 1-2 shows a two-story building framed using

cold-formed steel sections. In ta11 multistory buildings the main fkaming is

typically of heavy hot-rolled or welded sections, to support the primary

loads, while the secondary elements such as studs and joists (Fig 1-3) may

be of cold-forrned steel to support secondary loads.

Typical cold-formed panels are shown in Fig 1-4. These products are

generally used for such items as roofs, floor decks, wall panels, siding

material, and concrete forms.

1.2 BEHAVIOUR OF COLD FBRRIED MEMBERS

The current approach to the design of structures is based on a limiting stress

dictated by the proportions of the section. This limiting stress is related to the

mode of failure which may be governed by any of a number of factors.

Tension mernbers under static loads can fail by general yielding of the cross

section or rupture at the net section. Members in axial compression may

reach their maximum strength by yielding, by local buckling of thin

elements, or by overall buckling in flexure or torsion, or by a combination of

modes. Members in bending may be controlled by yielding, mpture at a net

section, local buckling or lateral buckling.

The study concentrates on two cornmon cold-formed sections which belong

to separate categories:

1. The behaviour of a box section is used to represent flat elements in formed

panels, and channel webs, where the elements are supported along both

long edges.

2. The channel section, in which the flange represents an element supported

along one long edge only.

In al1 cases of buckling, the critical elastic stress, O,, is given by:

where

E : elastic modulus A: slenderness ratio

= KL/r for flexural buckling in columns = mb/t for local buckling of plate elements

b: width K: a factor related to flexural buckling L : length m: a factor related to local buckling r: radius of gyration t: thickness

For an axially loaded column failing in flexure or torsion, the critical load is

the ultimate load and the theoretical forcelshortening relationship is as

shown in Fig 1-5 curve 1. In the case of flat elements supported along both

long edges which posses postbuckling strength, the theoretical force1

shortening relationship is shown in Fig 1-5 curve 2.

1.2.1 RECTANGULAR BOX COLUMNS

The ultimate axial compressive strength of a rectangular box colurnn is

govemed by either overall buckling or local buckling. Overall buckling leads

to collapse, but after initial local buckling of the flat elenients there is a

reserve of strength.

The initial stress to cause elastic local buckling is given by:

For a uniform axial stress, on a square box section, m=1.65 (CSA-S 136-94)

The dimensions used are shown in Fig 1-6.

Up to initial local buckling, the forcekhortening relationship is assurned to

be linear and the stress is assumed to be uniformly distributed over the cross

section. After buckling, the behaviour changes:

The relation between force and shortening is nonlinear

The distribution of stress over the section is not uniforrn but takes

the form shown in Fig 1-7.

The local deflection of a buckled element becomes large

The applied force can be increased until the yield strength is

reached at the corners of the section.

1.2-2 CHANNEL COLUMNS

Channels are open sections with only one axis of symmetry. In this case

failure may be by local buckling of the flanges, overall flexure or torsion, or

by a combination of these. Buckling in any of these modes precipitates

collapse.

The flange of an open section, such as in a charnel, is considered to be an

unstiffened element, supported along only one longitudinal edge.

Local buckling of the flange is given by equation (1.2) with a value of m

between 3 and 5 depending on the ratio of the flange width to the web depth.

If the flange buckles first, in a pin ended member, the section cannot cany

more load and this critical buckling load can be assumed to be the ultimate

load. However, if the web buckles first, the element c m cany more load, as

in a box section.

CHAPTER 2

THEORETICAL MODELS

In this chapter the current design methods, taken fiom CSA S 136-94 "Cold

formed steel structural members", which is typical of such Codes in general,

are introduced, followed by a description of the analytical mode1 used in this

study for plates with one or both longitudinal edges supported, representing

the flanges of channels and the walls of boxes.

2.1 OVERALL BUCKLING

For overall flexwal buckling, CSAS 136-94 uses a relationship between the

elastic critical stress, o;,, and the actual buckling stress, cc, given by:

When normalized this becomes:

where

The curve is shown in Fig 2-1. In this figure the CSA Code result for plate

and column are compared with this thesis theory and von Kannan's results.

2.2 BOX SECTION

2.2.1 CODE PROCEDURES

The manufacture of cold-formed section requires a radius at each corner.

(The form of radius differs for that in hot-rolled sections as shown in Fig 2-2)

In today's design codes the influence of this radius is to reduce the width to

the "flat width", which is used in design.

In a11 the current codes for strength design in cold-formed steel the following

procedure is adopted for walls with both long edges supported:

a For al1 proportions of the cross section, local buckling is treated by

considering the flat elements to be joined together at the corners

with simply supported edges.

Only the flat width (w) is considered to buckle locally.

Below a specified limit for the flat width (w') there is no buckling.

For greater widths, after initial local buckling, zones at the edges of

the flat width are assumed to carry the yield strength. (The sum of

these two zones is called the "effective width", b, )

Radiused corners are assumed to carry the yield strength, but are

otherwise disregarded

The critical buckling stress is given by (CSA S 136-94) :

Using Poisson's ratio, v=0.3, this gives:

A = 1.65(%)

The critical axial force for the flat width is then:

where

A = w t

To calculate the ultimate force, the Codes introduce an effective width

which can carry the yield strength, based on a formula proposed by Winter

(1 967), given below in its normalized fom (ASCE-8-90):

for 3 < 0.673, bfl = w

for n > 0.673, bg = p w

where:

The ultimate force on the "flat width" for a singular plate is then given by:

The four walls of a square box will thus resist a total ultimate force:

That the "flat width" should be used was the result of a cornmittee decision

based on an intuitive feel for behavior but with no support fiom theoretical

or experimental evidence.

This study undertalces to show the true influence of the radius, and, also, of

the influence of adjoining elements.

2.2.2 PROPOSED ANALYTICAL MODEL

To obtain theoretically the critical stress, a rectangular box with the

foilowing dimensions is considered:

a: shorter width of box b: longer width of box c: halfwave length of buckled element R: radius at the corner t: thickness

The box is assumed to be subjected to a uniform axial stress. The energy

method, (Timoshenko and Gere (1961)), in which the total strain energy of

distortion of the plates is equated to the work done by the axial stress in the

longitudinal direction as buckling occurs, is used to obtain the local initial

elastic critical stress.

The strain energy has two components:

m That due to deflection of the plates

r That due to twisting at the corners

The strain energy due to bending of the plate of width b, supported on al1

edges, can be written in terms of the deflections as follow: (Fig 2-3)

(Timoshenko, The terms related to twisting is equal zero in box sections)

where w is the deflection at the point (x,y) and D is the plate flexural rigidity

given by:

The strain energy due to twisting at the corners is calculated based on the

resistance to warping of the radiused portion.

The general form of strain energy due to warping is (Timoshenko) :

where H is the warping constant and:

The work done by the compressive extemal force as the elements shorten

during buckling is given by:

in which N is the load per unit width of the section.

The change in strain energy in the member is equal to the work done, thus:

U B + U H - T = O (2.18)

U, ,&,, and T are for the total section.

This is a minimum energy state. Differentiating equation (2.16) with respect

to the coefficients a, and a, and equating the results to zero, gives two

equations. The determinant of these two equations is theii equated to zero

and solved for N.

The deflected shape of each wall of the box is assumed to be given by an

expression of the form:

The cosine term represents the influence of the corner restraint. The

complete procedure to obtain the theoretical critical stress is presented in

Appendix 1.

2.2.3 THIEORETICAL POST-BUCKLING

The above analysis deals with elastic buckling and gives the critical stress.

After exceeding this stress the deflection increases rapidly, but the plate can

support the increasing load by a redistribution of the stress. The limiting

capacity is reached when the stress at the supported edges reaches the yield

strength. An approximate expression that determines the ultimate load is

represented below.

The theory assumes a perfect plate and linear material behaviour, but real

plates are not completely flat before loading and materials are nonlinear as

the yield strength is approached. The result is that the deflection increases

from the beginning of the load application, and buckling is not a sharp

action, furthermore the stress distribution is not unifonn fi-orn the beginning

of loading, with higher stress at the edges. The tme initial buckling stress,

cc, differs fkom the ideal elastic value, a,, and is given by a curve based on

experiments, of the forrn shown in Fig 2- 1, taken fkom CSA-S 136-M94 for

columns.

After buckling the stress distribution is shown in Fig 1-7 and the total load is

given by:

which is the area under the actual stress distribution curve. It can be seen that

in the limiting condition the plate may be approximated by two strips

carrying the yield strength as describe in 2.2.1.

In the general case, the width of plate, when the critical stress is equal to the

yield strength, is obtained fiom:

fiom which:

The total axial force on the flat plate is then oYb , r . Von Karman (1940)

proposed that this value of the axial force remained constant for al1 greater

values of b, thus when b>b, the mean axial stress at the ultimate load is:

From tests, this expression has been shown to be unconservative. It is

proposed to replace a, by the true buckling stress oc fiom the buckling

curve shown in Fig 2-4.

The normalized average stress can then be expressed by:

and the ultimate force is given by:

The resulting relationship is compared with the code formula [Winter

(1967)l in Fig 2-4

In this study it is proposed that in the general case the ultimate load is given

in which the buckling stress, O,, will be that for the element of the box,

taking into account the corner radius and the adjoining walls, and s is the

total length of the walls:

2.3 CHANNEL SECTION

2.3.1 CODE PROCEDURES

A channel section is considered to be an assembly of individual plates

simply supported at al1 edges except at the fiee edges of the flanges. In this

way the design is simplified to the design of one plate simply supported on

both long edges and two plates simply supported along one long edge and

fkee the other.

The current method used in Canadian codes is based on the effective areas of

these individual plates. The following procedure is adopted:

Only the flat widths (w) are considered to buckle locally.

The corners are treated as simple supports.

Below a specified limit for the flat width there is no buckling.

Q For greater widths, after initial local buckling, zones at the edges of

the flat width are assumed to carry the yield strength.

Radiused corners are assumed to carry the yield strength but are

othenvise disregarded.

0 The web of a channel is treated in the sarne manner as the elements

of a box section.

For flanges, called "unstiffened elements", the critical buckling

stress, neglecting any restraint by the web, is given by (ASCE-8-90)

Using Poisson's ratio, v=0.3, this gives:

2.3.2 PROPOSED ANALYTICAL MODEL

The following dimensions are used for obtaining theoretically the critical

stress of a channel:

a: flange width b: web width c: halfwave length of buckled element R: radius at the corner t: thickness

As with the box channel the energy method is used to obtain the critical

stress. The strain energy is supplied by:

a Deflection of the web, u,

Bending and twisting of the flange, U,

a Twisting at the corners, UT

The change in strain energy in the member is equal to the work done, thus

(Timoshenko) :

The complete procedure for finding the theoretical critical stress is presented

in Appendix 1.

Should the ratio of the web depth to the flange width exeed three, for

uniform stress, the web is less stable than the flange,and it is restrained by

the flanges. This is shown as the "Web Mode" inFig 2-5. For smaller ratios,

the flange is less stable and is restrained by the web shown as the ccFlange

Mode" in Fig 2-5.

2.3.4 THEORETICAL POST-BUCKLING

Up to this point the analysis has been concerned with the theoretical local

elastic buckling stress. In a concentrically loaded column, flange buckling

leads to collapse at the critical local buckling stress as there is no

rnechanism to develop postbuckling strength.

If the web buckles first it possesses postbuckling strength and the average

limiting strength for the web area based on the box section calculation,

becornes:

where o, i s the actual buckling stress for the web obtained fkom equation

2.2, using  for the web with m=1.65.

CHAPTER 3

FINITE ELElMENT STUDIES

3.1 INTRODUCTION

The more complex a section is, the more difficult is evaluation with

theoretical analysis. This is the reason that, today, many plate stability

problems in practical engineering are evaluated by numerical methods such

as the finite element method, which give approximate solutions for the ruiing

equations. In the finite element method the exact differential equation at a

point is replaced by an algebraic expression consisting of the value of the

function at that point and at adjacent points. In other words, the analytical

solution describes the behaviour of the systern as a continuous system, while

the finite element solution gives an discrete solution of the function. In

general if a continuous system is replaced by a discrete system the analytical

function is replaced by numerical values which represent the function at each

point.

In this study finite element methods provide a means to obtain an

independent check on the results of an analytical approach and on the

validity of code procedures. Box and charnel sections were analyzed using

the FEM program ABAQUS to obtain:

The critical elastic buckling stress using the Eigenvalue method

0 The ultimate, postbuckling capacity, using controlled displacement

3.2 EIGENVALUE METHOD

The eigenvalue represents the ideal elastic critical stress for initial buckling,

and is used here to compare the values obtained for different boundary

conditions. The formation of the eigenvalue problem is based on stability

analysis. The procedure may be stated as follows. Given a system with an

elastic stifkess matrix (K:: ), a loading pattern defined by the vector (eM ),

and an initial stress and load stifhess matrix (K"), find load multipliers

(A,), and buckling mode shapes @y), which satisfy:

in which N and M refer to degrees of the fieedom of whole system and i

refers to the ith mode. The critical buckling loads are then given by CZ,QM).

Usually only the smallest load multiplier and its associated mode shape are

of interest.

As an illustration

supported plate is

of the procedure of the eigenvalue method a simply

considered. Assume a square thin elastic plate, simply

supported on al1 four edges is loaded by an axial compression force in one

direction. (Fig 3-1)

The plate is divided into 20 elements (eight-node elements with 4 node on

the corner and 4 node on mid points). Due to symmetry, only a quarter of the

plate need to be considered. Briefly, the elastic sti&ess matrix[k,,], is

formed by adding al1 the element stiffness matrices which is given by:

in which [6] is the matrix of the deflection on 4 nodes of the element:

[4] , [a] are given by:

[Dl is the material constant matrix:

and N is the external work matrix on the element, given by:

Consequently to detennine the critical load it is necessary to obtain the non-

zero solution of:

{[k, 1 - 1% ImI = 0 (3.8)

in which [A] consists of the section nodal defiections and [Il is the index

matrix.

This is done by setting the determinant of the matrix equal to zero and

solving for the smallest root of the resulting equation.

3.3 ULTIMATE LOAD

To determine the collapse load for shells, a load -displacernent analysis is

performed. This checks that the eigenvalue buckling prediction is accurate

and at the sarne time investigates the behavior after buckiing, and up to the

collapse load.

The collapse load is defined as the maximum value in the load -

displacement curve. To find the maximum load, a stepwise increasing load

or displacement is applied. If the applied load is incremented it is called

"load control", and if the axial shortening is incremented it is ca-lled

"displacement control". Briefly, for "load control" an equilibrium system is

assumed to which a srnaIl increment of load is added. This unbalance causes

a displacement in the system which c m be represent as:

where u, , is the initial displacement and A U , is a small increment which can

be evaluated based on the incrernent in load.

This displacement becomes the initial displacement for the next step with a

new load incrernent. It is the sarne process for "displacement control" by

assuming steps in displacement.

3.4 ABAQUS PROGRAM

3.4.1 INTRODUCTION

The program which is used for numeric evaluation is ABAQUS. This

program is a general finite element program. It is an interactive preprocessor

used to create finite element models for shells. It has a variety of analysis

types. Those of the interested in this study are eigenvalue extraction and

static stress analysis, which are used for pre- and post-buckling. The program

accepts the input which includes:

Mode1 Data

History Data

The data to define a finite element model are:

the elements: number, properties

nodes: number

material properties

History Data represents the initial condition or sequence of events or loading

for the model. History data for eigenvalue method is basically an arbitrary

initial load based on this method, and has only one step. For load -

displacement analysis it represents the properties at each step which include

both load and displacement.

3.4.2 INPUT OF THE PROGRAM.

For executing the program, based on the method used, the following input is

required:

DEFINING OF MODEL:

Elements and Nodes: The number and type of element (classified

by the number of nodes in each element

Shell: The elernents in each shell and the nurnber of layer in each

shell with their thicknesses.

Material: The elastic modulus, yield strength and Poisson ratio for

each layer of the shell

Boundary and Symmetric Condition: The conditions on each

boundary and any syrnmetry

DEFINING OF HISTORICAL DATA

Eigenvalue method: An initial stress distribution on the specific

boundary is required. (Appendix III- List 1)

Load - Displacement analysis : In case of Load Control an initial

and a step increment stress distribution on the required elements.

For the case of Displacement Control an initial and a step increment

strain distribution on the required elements. (Appendix III- List 2)

3.4.3 OUTPUT OF THE PROGRAM:

For each case by executing the eigenvalue section of the program, ABAQUS

predicts the buckling modes and corresponding eigenvalue. Also a list of

stresses and deflections is provided in the output of the program.

The load-displacement analysis results for each case are typically provided

in n steps fiom zero load to a maximum which is part of the input. At each

step the stress and strain distribution over the section is provided in the

output.

CHAPTER 4 COIMPARING THE RESULTS

The numerical, finite element results are compared with the theoretical

predictions and the Codes rules. The fhite elernent study was made for box

and channel sections in the following ranges:

BOX SECTION RANGE

Ratio of widths, a h (a < b): 0.25,0.5, 1 Ratio of width / thickness, b/t 40, 50, 66, 100, 200 Ratio of corner radius / width, R h O to a/2

CIXANNICL SECTION

Ratio of web / flange width, ah 1, 2, 4 Ratio of flange width / thickness, b/t 40, 50, 66, 100, 200 Ratio of corner radius / flange width, Rh O to b/2

4.1 TYPICAL EXAMPLE

To show the procedure for each case the following typical box section is

chosen (Fig 4-1):

4.1.1 'l'TE0RETICA.L CALCULATION

D is given by:

Warping factor a=0.044, and the warping constant H is given by (see appendix II for details):

The elastic critical stress obtained fiom the theoretical analysis is:

The norrnalized slenderness is:

Thus, - according to the CSA standard, the tme buckling stress for h > Vd0.6 = 1.29, is:

The predicted mean stress at collapse, according to the method proposed in this thesis (Equation 2.24) is then:

The true length of the walls is:

Then the collapse load is:

4.1.2 CODE PREDICTIONS

The flat width is :

w = b - 2 R = 1 0 0 - 2 x 5 = 9 0 mm

giving a slendemess of :

f = 1 . 6 5 % = 1 . 6 5 ~ 9 0 ( = 1 4 9

The critical buckling stress is given by:

Leading to:

and:

The ultimate load fiom the Code is given by:

Thus the predicted average stress at collapse is:

4.1.3 FINITE ELEMïENT RESULTS

4.1.3.1 EIGENVALUE RES'ILTLT

The eigenvalue is detennined using load-control. Based on the input shown

in Appendix III-List. 1 the elastic critical stress for this case is (Fig 4.2):

4.1.3.2 POSTBUCKLING STRENGTH

The ultimate load is determined using displacement contro1,Appendix III-

List.2 shows the input format. At each step, the stress distribution across the

boundary element set is obtained, and the average stress is evaluated. For the

typical example being demonstrated, the stress distribution across the section

at the ultimate load is shown in Fig 4.3.

Fig 4.4 shows the relationship between the mean stress and axial shortening.

The highest stress in this curve represents the ultimate average stress,

which is:

The ultimate force is given by:

F~;""~*' = a ; l b lSAPUSst =162~391~1=63500N

4.2 RESULTS FOR BOX SECTIONS

The results are given in the figures listed in following table:

Fig 4.5 Fig 4.6 Fig 4.7 Fig 4.8 Fig 4.9 Fig 4.10 Fig 4.1 1 Fig 4.12 Fig 4.13 Fig 4.14 Fig 4.15 Fig 4.16 Fig 4.17 Fig 4.18 Fig 4.19

Fig 4.20 Fig 4.21 Fig 4.22 Fig 4.23 Fig 4.24 Fig 4.25 Fig 4.26 Fig 4.27 Fig 4.28 Fig 4.29 Fig 4.30 Fig 4.3 1 Fig 4.32 Fig 4.33 Fig 4.34

Critical Critical Critical Critical Critical Critical Critical Critical Critical Critical Critical Critical Critical Critical Critical

Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate ültirnate Ultimate

In ench figure the Code, theory and ABAQUS result for critical and ultirnate stresses are shown. Table 4-1 shows the summary of the results for al1 the cases. In this table how the Code and the theory differ fiom ABAQUS is shown as a percentage error. The m factors used in equation 1.1 are represented in Fig 4-35 to 4-37, and values normalized with respect to 1.65, for box sections, are shown in Fig 4-3 8 to 4-40, as they Vary with Rh.

Finally following figure shows the normalized mean stress at the ultirnate load for theory, ABAQUS and Code predictions.

Normalized column buckling stress

1

0.8 0 ABAQUS O - Code

- Theory

The extreme cases based on the percentage error between Code and ABAQUS result are s h o w in following table:

a h

1 0.25

4.3 RESULTS FOR CHANNEL SECTIONS

The results are given in the figures listed in following table:

a/b

1

Fig 4.42 Fig 4.43 Fig 4.44 Fig 4.45 Fig 4.46

Fig 4.47 Fig 4.48 Fig 4.49 Fig 4.50 Fig 4.5 1

Fig 4.52 Fig 4.53 Fig 4.54 Fig 4.55 Fig 4.56

b/t

40 200

Critical Critical Critical Critical Critical

Critical Critical Critical Critical Critical

Critical Critical Critical Critical Critical

b/t

200

R/b

0.3 0.1

0.25 1 200

Rh

0.3 0.125

Critical wmm2) stress

Code

Error -79% 65%

Code 2832 28

Ultimate (N/mmz) stress

Theory

Emor 9% 11%

Code

Error -20%

Code 174 106

Theory 1435 72

Theory

Error 7%

40%

ABAQUS 1578 81

10%

Theory 134 160

ABAQUS 145 177

In each figure the Code, theory and ABAQUS predictions for critical stresses are shown. Table 4-2 shows the summary of the results for al1 the cases. In this table how the Code and the theory differ fiom ABAQUS is shown as percentage errors . The m factors are represented in Fig 4-57 to 4-59 and normalized values with respect to 5 for the flanges, are shown in Fig 4-60 to 4-62 as they Vary with Wb. Finally Fig 4.41 shows the normalized mean stress ai the ultimate load for theory, ABAQUS and Code results.

The extreme cases based on the percentage error between Code and ABAQUS result are shown in following table:

1 a h ( b/t ( Rh 1 Critical (N/mm2) 1 Code Theory L 1 200 0.1 Error 78%

stress ABAQUS 74

Code 16

Theory 69

4.4 CONCLUSIONS

4.4.1 BOX SECTIONS

The influence that the radii at the corners has on the elastic and ultimate

buckling stress of the box sections is important and should not be neglected,

as in the current codes.

For boxes with a low width to thickness ratio, 66 or less, failure occurred

close to the yield strength. In these situations the buckling stress is more

influenced by the ratio of the widths, (ah) than Wb. For a high width to

thickness ratio, more than 66 when the failure stress is lower than yield

strength, both the ratio of the widths and R h are very influential on the

failure. By neglecting the bla ratio and R h the Code c m be 65%

consenrative to 79% unsafe.

4.4.2 CHANNEL SECTIONS

The influence that the ratio of the web to flange, bla, has on the buckling

capacity of the section is important and must be considered. For the channels

with a web to flange ratio less than 3 failure occurred by flange buckling

mode, and the critical stress represents the ultimate stress. For a web to

fiange ratio more than 3 web buckling occurred first. In this case there is a

possibility for the channel to carry an ultimate load in exceed of critical

elastic value. By neglecting the b/a ratio the Code can be varied fiom over

34% to 54% conservative while by neglecting Wb, the Code canbe 78%

consemative.

4.4.3 FINAL COMMENTS

An inappropriate interpretation of the ratio of the width for the cold-formed

sections and the neglect of the influence of the corner radii on the buckling

capacity as in the current codes, c m lead to unsafe design or overdesign.

4.4.4 FUTURE RESEARCH

To follow the present work, which is just done for two major forms of cold-

formed section (box, charnel) it is recommended that other be investigated.

APPENDIX I

ANALYSE

1.1 INTRODUCTION

In this chapter the steps of the procedure are described. The software

which is used for parametric calculation is Mathcad which has the ability

to solve algebraic equations. To obtain the result for each specific case

the numerical values of the parameters are substituted in the final

algebraic expressions.

1.2 BOX SECTION

After buckling the deflection of the elements is assumed to be rnodeled

by trigonometric expressions as follows: (shown in Fig 1.1)

where a,.a,,a,,a, define the buckled shape and a < b.

As it was explained in Chapter 2 the strain energy due to deflection in a plate supported on al1 edges, is given by :

Considering the affect of the corner radii on the width of the plates for side b the strain energy due to bending is given by:

in which p is given by:

P(+

and for side a it is given by:

. +4&a.a 3.a4.c4.EOi[fk$-] +48.. 4 2 . c ~ . m s [ 1 . w ( O a ] . r i n [ Z n q ] - a

The external work due to uniform axial force given by:

The external work for side b is given by :

The warping constant is given by:

Where is the net displacement normal to a cross section due to

warping and the length s is measured along the element wall.

(For evaluation of H see Appendix I I )

The strain energy due to warping for the corner between sides a and b is given by:

in which

This leads to:

The energy balance for the whole section is given by:

U a i - U b f U ~ - ( r a + T b ) a (1-1 4)

This equation has five unknowns, NSa 1.a2,83.a4. At each corner the

tangents to the deflected plates must be perpendicular to each other.

This provides the third equationwhich gives a, in terrns of a l :

" 1.; (1-1 5 )

Considering the bending moment for sides a and b, to provide equal

moments at the corner for both ' plates the following equation must be

valid:

Substituting the value of a, and ., in term of , leads to an equation

with three parameters. Differentiating with respect to ., and ., gives hivo

equations. To find the theoretical critical stress the determinant of these

equations is equated to zero.

Differentiating with respect to . , gives:

F 1 l(N1-a 1 +F 12(N).a Zt30

Differentiating with respect to ., gives:

Equating the determinant of equations 5-17 and 5-18 to zero leads to:

The critical theoretical buckling force is evaluated from F , l(N)

to F 22(N) as:

1.3 CHANNEL SECTION

For a channel section the procedure is divided into two phases.

Depending on the ratio of web width to Range width, the critical buckling

stress for the web may be srnaller or larger than the critical stress for the

flange. The true buckling stress is that of the combined section, but the

buckling mode rnay be described as "flange mode" or "web mode".

1.3.1 FiANGE MODE

Assuming the flange buckles first, the deflection of the elements after

buckling is given by: (shown in Fig 1.2)

Web :

~ X Y w webs[a sin(";) + s 2. (1 - oos(+))].sin(nt)

Flange:

flangr=[o-X + a (1 - cos(z))]-rinjw:)

where e , a l , a 2 , a 2 define the buckled shape.

The strain energy due to bending for the web is given by:

where:

For the flange the strain energy is given by:

The external work for the web is given by:

and for flange it is given by:

The strain energy due to twisting at one corner is given by:

in which

This leads to:

The energy balance for the whole section is given by:

web + 2.U fiange + H- (T web + 2.T flange)'* (1-35)

This equation has five unknowns N,B,a ,,a,,a:. At each corner the tangent

to the plates must be perpendicular to each other. Thus:

X B a 1 s b

Equal bending moment at the corner for both plate gives:

Finally by substituting the value of a 3 and 0 in term of a ,,a, ,leads to an

equation with three parameters and the rest of the procedure is the same

as that for the box section.

1.3.2 WEB MODE

Assuming the web buckles first, the deflection of the elements after buckling

is given by: (shown in Fig 1.3)

Web :

where e , a l , a , , a l define the buckled shape.

The strain energy due to bending for the web is given by:

and for flange it is given by:

The external work for the web is given by :

and for flange it is given by:

The strain energy due to warping for a corner is given by:

Using the equations 1-41 and 1-44 and substituting the values of a, and

0 in term of al,a2 leads to an equation with three parameters, and the

rest of the procedure is the sarne as that for the box section.

APPENDIX II

WARPING CONSTANT

Each radiused corner of a box or channel section possessesa

warping constant which depends on the geometry of the section.

Based on the rnethod of ~imoshenko and Gere (1 961) the following

procedure is used to calculate the warping constant.

11.1 BOX SECTION

~ h e corner is opening up as shown in Fig 11.1. This is then treated

as a beam "loaded" by the normal distance from the intersection of

the median lines of the adjoining plates to the tangent to the corner

element, as shown in Fig 11.2. This normal distance is given by:

I=R- (sine + cos^ - 1) (11-1)

The total "load" is given by:

R and R , the reactions of the ends of the "beam", as shown in

Fig 11.2, are:

II- 1

The "shear force" in the beam is shown in Fig 11.2. This shear

force represents the term of (5,- a$ in the warping constant equation: . - - - - - - - - - - - . A -- -- - - - - - - -

The evaluation of this integral is based on two parts which are:

4 . lntegration on the straight parts

2. lntegration on the curve part

and H is given by:

The warping constant H can be expressed by (Fig 11.3):

in which:

b= longer length CL= a factor that varies with a/b and R/b shown in Fig 11.4

The value of a is given closely by:

11.2 CHANNEL SECTION

The corner is opened up as shown in Fig 11.5. The procedure is the sarne

as that for the box; however, in the channel section the centre of rotation

for the corner is no longer at the intersection of t h e median lines but at

a distance e from this point, on the rnedian line of the web, and is

- - - - - - - -- - - - - - - - - - - -establisried by niininiiting -the- YvaFpiig consfit as Toll6ws: -

The normal distance from the centre of rotation to t h e element tangent is

g iven by (Fig 11.6):

The total "load" is given by:

R, and R ~ , the reactions of the ends of the open corner beam, as shown

in Fig 11.6, are calculated using:

and

thus:

- - - - - - - A . - . . - - - - - -

force in the beam, as shown in Fig 11.6 represents the term of

the warping constant given by:

The evaluation of this integral is based on three parts which are (Fig 11-6):

i. lntegration on C part

2. lntegration on D part

3. lntegration on F part

1 (R 1 + e . ~ ) ~ dr P=.[(R 1 + e-b - e - ~ ) ~ - R 13]

O

where:

Lab-R

Differentiating the warping constant with respect to e, and equating it to _ _ _ _ _ _ _ _ _ - _ _ _ -_. _- _ - - - - - ---- - - - - - - - - - - - -

zero gives:

The warping constant H can be expressed by (Fig 11.7):

in which:

b= longer length a= a factor that varies with alb and Rlb shown in Fig 11.8

The value of a is given closely by:

APPENDIX III LIST-1 *HEADING SQUARE PLATE - ELASTIC BUDKLING ( BUCKLE )-S8R5 2 X 2 4-2-3-1 *restart, write *NODE 1010 1018,0.0,0.45,0.0 lOZ!4,O.O5,O.5,O.O 1032,0.5,0.5,0.0 2010,0.0,0.0,3.0 2018,0.0,0.45,3.0 2024,0.05,0.5,3.0 2032,0.5,0.5,3.0 1,0.450,. 050,O.O 2,0.450t .050,3.0 *NGEN, NSET=ZYEDGl 1010,1018,1 "NGEN, NSET=ZYEDG2 2010,2018,l *NGEN,NSET=ZXEDGl 1024,1032,l *NGEN, NSET=ZXEDGS 2024,2032,l *NGEN,NSET=ZRl,LINE=C 1018,1024,1,1 *NGENtNSET=ZR2,LINE=C 2018,2024,1,2 *NGEN,NSET=YZEDG 1010,2010,100 *NGEN, NSET=XZEDG 1030t2030, 100 *NFILL ZYEDGl,ZYEDG2,10, 100 *NFILL ZXEDGl,ZXEDG2,10,100 "NFILL ZRl,ZR2, 10,100 *ELEMENT, TYPE=S8R5, ELSET=ONE 1,1010,1012,1212, 1210, 1011, 1112, 1211, 1110 *ELGEN, ELSET=ONE 1, 11, 2, 1, 5,200, 11 "MATERIAL ,NAME=PLATE "ELASTIC 20.E4, . 3 *SHELL SECTION,MATERIAL=PLATE,ELSET=ONE .Olt 3 * BOUNDARY ZYEDG1,l ZYEDG1,Q ZYEDG1,G ZYEDG2,l ZYEDG2,4 ZYEDG2, 6 ZXEDG1,2 ZXEDG1, 5 , 6 ZXEDG2,S

ZXEDG2,5,6 ZR1,1,2 ZR1,4 ZR11 6 ZR2,1,2 ZR2,4 ZR1,6 YZEDG, YSYMM *STEP *BUCKLE 1,,,99 *MODAL FILE *CLOAD 1010t3,1.458 1011,3,5.833 1012,3,2.917 1013,3,5.833 1014,3,2.917 lOl5,3,S. 833 1016,3,2.917 lO17,3,5.833 1018,3,2.767 lOl9,3,5.236 1020,3, 2.618 1021,3,5.236 1022,3,2.618 1023,3,5.236 1024,3,1.726 1025,3,1.667 1026,3, O. 833 1027,3,1.667 1028,3, O. 833 lO29,3,l. 667 1030,3,0.833 1031,3,1.667 lO32,3, 0.417 2010,3, -1.458 2011,3, -5.833 2012,3, -2.917 2Ol3,3, -5.833 2Ol4,3, -2.917 2Ol5,3, -5.833 2Ol6,3, -2.917 2Ol7,3, -5.833 SOl8,3, -2.767 2019,3, -5.236 2020,3, -2.618 2021,3, -5.236 2022,3, -2.618 2O23,3, -5.236 2024,3,-1.726 2025,3, -1.667 2026,3, -0.833 2O27,3, -1.667 2O28,3, -0.833 2029,3, -1.667

2O3O,3, -0.833 2031,3, -1.667 SO32,3, -0.417 *NODE PRINT u *END S T E P

APPEHDIX III LIST-2 *HEADING SQUARE PLATE - RIKS STEP ANALYSIS ( STATIC )-S8R5 2 X 2 4-2-3-1 "restart, write *NODE 1010 1018,0.0,0.45,0.0 1024,0.05,0.5,0.0 1032,0.5,0.5,0.0 2010,0.0,0.0,3.0 2018,0.0,0.45,3.0 2024,0.05,0.5,3.0 2032,0.5,0.5001,3. O 1,0.450, .050,0.0 2,0.450, .050,3.0 *NGEN,NSET=ZYEDGl 1010,1018,1 "NGEN, NSET=ZYEDG2 2010,2018,1 *NGEN,NSET=ZXEDGl 1024,1032, 1 'NGEN, NSET=ZXEDG2 2024,2032,l "NGEN, NSET-ZR1, LINE=C 1018,1024,1,1 *NGEN, NSET=ZR2, LINE=C 2Ol8,2O24,l, 2 *NGEN,NSET=YZEDG 1010,2010,100 *NGEN, NSET=XZEDG 1030,2030,100 *NFILL ZYEDG1, ZYEDG2, 10,100 *NFILL ZXEDG1 , ZXEDG2,lOt 100 "NFILL ZRl,ZR2, 10,100 *ELEMENT, TYPE=S8R5, ELSET=ONE 1,1010,1012,1212, 1210, 1011, 1112, 1211, 1110 *ELGEN, ELSET=ONE 1, 11,2, 1, 5, 200, 11 *MATERIAL , NAME=PLATE "ELASTIC 20.E4, .3 *SHELL SECTION,MATERIAL=PLATE,ELSET=ONE .Olt 3 "BOUNDARY ZYEDG1, 1 ZYEDG1, 6 ZYEDGS, ZSYMM ZXEDG1, 2 ZXEDGl, 6 ZXEDG2, ZSYMM ZR1,1,2 ZR1,6 ZR2, ZSYMM YZEDG, YSYMM *STEP, NLGEOM, INC=400 "STATIC, RIKS t t t t1030t3t *2

*MODAL FILE "CLOAD 1010,3,. 0166666 1011,3, .O666666 1012,3, .O333333 1013,3,. 0666666 1014 13, -0333333 1015,3, ,0666666 1016,3,. 0333333 lOl7,3,. 0666666 1018,3, .O333333 1019,3,. 0666666 1020,3,. 0333333 1021,3, .O666666 1022,3,. 0333333 1023,3, .O666666 1024,3, .O333333 1025,3, .O666666 lO26,3, .O333333 1027,3, .O666666 1028,3, .O333333 1029,3, ,0666666 103013,. 0166666 *PRINT,RESIDUAL=NO *NODE filE, NSET=N2030 U *END STEP

Original flat shape

shape

Fig D. 1 Local buckling of a box section

FD- 1

Original flat shape

/ Deflected shape

Fig D-2 Local buckling for a channel section

FD-2

Overal width b, and flat width w

Effective width, b,,

Fig D-3 Flat, overall and effective widths

shape

Fig D-4 Collapse mode for a box section

FD-4

Twisting without restrain

Twisting with fixed end - - - - - - - - - - - - - - - - - - - - -

Fig D-5 Warping for 1 beam

Fig 1 - 1 Cold-fonned sections

FI-1

Fig 1-2 Cold-formed steel structure (Pen Meta1 Company)

FI-2

Fig 1-3 Cold-fomed and hot-rolled steel structure (Stran-Steel Corporation)

FI-3

Foor and roof panels

Long-span roof panls

Floor and roof panels

c - - Curtain wall panels Ribbed panels Comigatec panels

Fig 1-4 Typical cold-formed panels

FI-4

Fig 1-6 Box and panel section dimensions

FI-6

Variation of o, at plate edge

Fig 1.7 Stress distribution on a simply supported plate

Hot Rolled Section

Cold-Formed Section

Fig 2-2 Cold-formed and hot-rolled corners

Flange buckling mode

Web buckling mode

Fig 2-5 Buckhg modes of a channel

Fig 3 - 1 Sample plate for ABAQUS

F3-1

Fig 4-1 Typical box section

Fig 4-2 Typical ABAQUS elastic buckling

F4-2

E = 200000 ~lrnrn '

criticai-stress = 86 ~ l m m ~

ultimate-stress = 165 ~ l m d

Fig 4-4 Mean stress and axial shortening for each step for a box section

Fig 4- 19 Critical stress for a box section, &=OS blt=200

- TïIEORY --&--ABAGIUS --- CODE

[I) . * S m s u - h ~ $ I I * ,

c d :

i l : :

=? Tl-

Fig 4-46 Critical stress for a charme1 section, a/b=l b/t=200

Fig 4-63 Normalized mean strength for channel sections

T h e o r y 0 ABAQUS

N N X-Y-Z 45 degree view

N Down X axis Down Z axis

M M X-Y-Z 45 degree view

M Down Y axis D o m Z axis

Fig 1-1 Deflection of a box section

Flange:

N N N X-Y-Z 45 de- view Down X axis Down Z axis

Web :

M M M X-Y-Z 45 degree view Down Y axis Down Z axis

Fig 1-2 Deflection of a channel section (flange mode)

Flange:

N N X-Y-Z 45 degree view Down X axis

Web :

N Down Z axis

M M M X-Y-Z 45 degree view Down Y axis Down Z axis

Fig 1-3 Deflection of a channel section (web mode)

Fig II- 1 Corner of a box section

Loading distribution 1 R2

Shear force distribution

FigII-2 Warping constant construction for the corrner o f a box setion

Y +

Fig 11-5 Corner of a channel section

Loading distribution

Shear force distribution

Fig 11-6 Warping constant construction for the corner of a charnel section

Fig II-7 Warping constant for channel sections ( t-1 )

9000000 --

8000000 --

7000000 --

6000000 --

H 5000000 --

4000000 -- - aib=l design formula

3000000 -- - alb=0.75 - alb=O.5

2000000 -- X a/b=l by analysis U alb=0.75

1000000 -- a ahO.5

O O 0.05 0.3 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Rlb

Table 4-1 Summary of results for box sections

Cc)

9 7

Table 4- 1 Sumrnary of results for box sections

Table 4-1 Surnrnary of results for box sections

Table 4- 1 Summary of results for box sections

Table 4- 1 Surnmary of results for box sections

Table 4- 1 Summary of results for box sections

Table 4-1 Sumrnary of results for box sections

Table 4- 1 Summary of results for box sections

Table 4-1 Summary of results for box sections

Table 4-1 Summary of results for box sections

Table 4-1 Summary of results for box sections

Table 4-2 Summary of results for channel sections

Table 4-2 Summary of results for channel sections

Table 4-2 Summary of results for channel sections

Table 4-2 Summary of results for channel sections

Table 4-2 Summary of results for channel sections

Table 4-2 Summary of results for channel sections

ABAQUS/STANDARD (Versi0n5.5)~ "Manual and Examples" Hibbit Karlsson & Sorensen, Inc. Email : hksGiWs.com

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